Let \(\alpha \in . In this subsection we discuss the differential operator\begin{equation}A_{\alpha } = x^{\alpha } \partial _x x^{- \alpha } = \partial _x -\frac{ \alpha }{x}\end{equation}acting on scalar functions. It will be a building block of somespecial 1d Dirac-Coulomb operators considered in subsections\ref {app:elementary}and \ref {app.super}.\) Let us briefly recall basic results about realizations of
\(A_\alpha \)  as a closed operator in \(L^2(\mathbb {R}_+)\)  following
[1]}. Proofs of all statements stated in this subsection
without justification can be found therein.
(In [1]} a different convention was used:
\(A_{\alpha } = -\mathrm {i}\partial _x +\frac{\mathrm {i}\alpha }{x}\) . Thus
\(A_\alpha ^{\rm new}=\mathrm {i}A_\alpha ^{\rm old}\) .)
